A SIMPLE PRECIPITATION MODEL 1
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1 Wer Reure, Hydrg nd Enrnmen Sene Dn C Engneerng Dermen Fr Cn, CO ( CIVE520 PHYSICAL HYDROLOGY A SIMPLE PRECIPIAION MODEL he ren mde reened n bed n he -ed db re dynm. M r re end fwng n db re he ee f ndenn, nd eud-db re be he ee f ndenn. he rdu f he ndenn re, qud wer dre, nrbue he frmn f he ud yem. Eenuy, f hee dre grw uffeny h her ermn ee exeed he udrf ey f he r, hey w f u f he ud nd eny mke he urfe ren. he m f qud wer nen f he ud yem ee n me rdng he fwng nnuy equn, dx ( I( O ( Ob ( ( d where X( he m f qud wer n he un ud umn, I( he nu fux f qud wer reu f ndenn fwng n en derbed erer, O ( nd O b ( re he uu fuxe f qud wer dre hrugh he ud nd ud bm reeey. he nnuy equn be n be ed n fne dfferene frm, X ( + Δ X ( + [ I( O ( Ob ( ] Δ (2 Exren mue f he erm n h equn re gen bew. he rbem emen rde f he neery n ue h h equn n be rged n me. Fr he fr me e, mue X(0, I(0, O (0, O b (0 ung equn 3, 4, 20, nd 2 bew nd ung he n ue gen n he rbem emen. Fr he end nd er me e, bn he ue f he nere men dmeer he rrendng me e by ung equn 8 bew, nd hen reed eue he ne fuxe nded. Fr furher de r mre mex ern f h mde, he reder referred Gergkk nd Br, 984, 984b; nd Br, 990. JORGE A. RAMIREZ
2 MASS OF LIQUID WAER IN HE CLOUD SYSEM he m f qud wer er un re n ud umn n be mued ung he fwng equn, X( (πρ w N Z δ / 4 (3 δ 3 ( γ + γ 2 + γ 3 (3b where N he rmeer f he Mrh-Pmer drbun f dre ze nd he nere f he men f he dmeer f he dre he ud bm; Z he ud deh; nd γ he r f he men ud dmeer he ud bm h he (yy >. he Mrh-Pmer drbun f ud dre ze ( D N e D η where η(ddd he number f ud dre wh dmeer n he rnge [D, D + dd] er un ume f ud. INPU MASS FLUX: In rder eue he nu f qud wer m he ud yem, we need eue he fwng equn. I ρ d Δw ρ d [w w (, ] (4 Fr h, we need bn he r deny. In ddn, we need mue he ne hnge n wer r mxng r fr re undergng n db-eud-db en frm he urfe he ud. he e neery d re uned bew. BELOW LFC: Fr m r endng dby beween he urfe nd he fng ndenn ee (LCL, he en emerure, θ, nd wer r mng r, w, re nered qune. hu, he LCL: θ ( Rd C nn (5 w εe ( nn (6 Lw (, θ e θ ex[ ] (7 C 2 JORGE A. RAMIREZ
3 In ddn, fr n db en, he emerure,, nd reure,, f he r re he LCL re reed hrugh he fwng equn: 00000( Pung equn 5-8 geher ed, θ C C n( θ e L θ εθ C R d e ( C R d Rd (8 where θ nd θ e re knwn ue (gen n he rbem emen nd nn bew LCL. Equn 9 mu be ed erey fr he re emerure LCL,. Afer bnng, he reure he LCL,, n be bned ung equn 8; he n mxng r, w, n be bned frm equn 7. Fny, he ude f he ud be n be bned, Z ( mb (0 b Γmb where ( mb he mben reure he LCL, whh n be mued eung equn 6 bew he LCL reure. ABOVE LCL: Fr eud-db en be he LCL, he equen en emerure nered, θ e, where he en emerure, θ, n. hu, he ud, θ e θ ex[ L w (, ] ( Rd C ex[ L w (, ] ( C C h equn funn f he emerure,, nd he reure,, f he re he ud. Hweer, he reure f he ud gen n he rbem emen. hu, equn funn f ny ne unknwn, nmey,, whh n be bned ung n ere rh. One bned, he fn wer r mxng r he ud n be bned frm, (9 w (, εe ( (2 Fny, Δ w [ w w (, ] (3 3 JORGE A. RAMIREZ
4 Fr he ure f h hmewrk ny, ue he deny f he r he urfe nd he udrf ey gen n he rbem emen n he equn fr I be. e ( R ρ m (4 d he mben r emerure he ud, ( mb, n be bned ung he reu bned n Hmewrk N. fr he drbun f reure n n mhere n whh he emerure deree nery wh een, g R d Γ mb ( z ( z z (5 ( z Eung equn 5 he ud fr whh (z nd (z nd ng fr he mben r emerure he ud, (z ( mb, Rd Γmb g ( mb ( whh n hen be ued bn he ude f he ud, (6 Z ( Γ mb mb (7 OUPU MASS FLUX: he uu m fux med f w nrbun, he uu f ud dre frm he f he ud, O, nd he uu f ud dre frm he bm f he ud, O b. Bh f hee fuxe re funn f he nere f he men f he dmeer f he un f ud dre he ud nd ud bm, reeey. he nere f he men dmeer n be bned frm he fwng equn: πρ N Z δ X w 4 [ ] (8 where X he m f qud wer nned n he ud umn nd Z he ud deh, Z Z Zb (9 4 JORGE A. RAMIREZ
5 OUPU FROM CLOUD OP: Cmue he uu frm he ud ung he fwng equn, O πρ N w α ( ( γ 5 6 [Γ(5,γ α γ α Γ(4,γ α ]+ γ 4 (20 α OUPU FROM CLOUD BOOM: Cmue he uu frm he ud bm ung he fwng equn, O b πρwn α [ Γ(5, Γ(4, ] 6 4 α α (2 where α he rmeer f he ermn ey funn, nd he udrf ey. In hee equn, eeryhng knwn he n me gen n he rbem emen. Fr her me e, he nere f he ud dmeer hnge reu f hnge n he m f qud wer nned n he ud umn. hu, he nere f he men dmeer hud be mued he begnnng f eh me e ung equn 8 be, n whh X eued he begnnng f he ner. In equn 20 nd 2, Γ(.,. nd fr he Cmemenry Gmm funn, defned n he endx. PRECIPIAION RAE: Cud dre fng u frm he ud bm my eenuy reh he grund nd nrbue he ren re. Hweer, fr unured ub-ud yer ndn, he m f he dre redued by ern f he grund. kng n un h re, he ren re he urfe n be mued ung he fwng equn, πρwn α 3 P [ Γ(5, (4, 4 D Γ D Dα 6 2 Γ(2, 3 D + D (22 where D hrehd dmeer defnng he drbun f dre he ud be h uy nrbue he ren re he urfe. Dre wh mer dmeer w eher n he ermn ee h exeed he udrf ey he ud be, r ere meey befre rehng he urfe. D mx( D, / α (23 In he be exren, D rereen he r dmeer ud be. Cud dre f h dmeer w meey ere exy un rehng he grund urfe. 3 e D ] 5 JORGE A. RAMIREZ
6 hu, ny dre whe dmeer ud be rger hn D w nrbue he ren re. Fr edy e Fkn dffun, he r dmeer n be bned, * 4D e ( w e ( d / 3 D [ Z ( ] * b R w (24 where * nn wh un f kg/m 3 / whh ke n he fwng ue, 7. x 0.4 x 0 rn 5 * 5 (25 nw nd D * he dffuy f wer r n r n m 2 /, whh funn f emerure nd reure nd n be mued, D * *0 ( ( ( In equn 24, he bue humdy he urfe f he dre h rrendng he we-bub emerure, w, he urfe ndn, nd he mben bue humdy h rrendng he dew-n emerure, d, he urfe ndn. ENERGY GENERAED PER UNI MASS he energy genered er un m reu f he wrk dne by he e buyny fre ng n he re beween he ee f free nen (LFC nd he ud n be bned ung he fwng numer rxmn (rezd rue he energy negr, Energy / M where, nd, ( 2 Cud LFC g( + dz + Δ z z+ + z + g n Δz (27 6 JORGE A. RAMIREZ
7 APPENDIX he nmee gmm funn requred n he be mun defned, Γ(ν, x x ν e d Fr neger ue f ν (whh he e fr h hmewrk, he nmee gmm funn n be eued ung he fwng exren, Γ(n, x (n!e x n k x k (k! (n!e x (+ x + x2 xn ! (n! REFERENCES Br, R. L., 990: Hydrgy, An Inrdun Hydrg Sene.Addn-Weey Gergkk, K.P. nd R.L. Br, 984: A Hydrgy Uefue Sn Pren Mde:, Frmun. Wer Reure Reerh. 20(, Gergkk, K.P. nd R.L. Br, 984b: A Hydrgy Uefue Sn Pren Mde: 2, Ce Sude. Wer Reure Reerh. 20(, JORGE A. RAMIREZ
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